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: <math>\mathbf{Y} = \mathbf{X}\mathbf{B} + \mathbf{U},</math>
where '''Y''' is a [[matrix (mathematics)|matrix]] with series of multivariate measurements (each column being a set of measurements on one of the [[dependent variable]]s), '''X''' is a matrix of observations on [[independent variable]]s that might be a [[design matrix]] (each column being a set of observations on one of the independent variables), '''B''' is a matrix containing parameters that are usually to be estimated and '''U''' is a matrix containing [[errors and residuals in statistics|errors]] (noise).
The errors are usually assumed to be uncorrelated across measurements, and follow a [[multivariate normal distribution]]. If the errors do not follow a multivariate normal distribution, [[generalized linear model]]s may be used to relax assumptions about '''Y''' and '''U'''.
The general linear model incorporates a number of different statistical models: [[ANOVA]], [[ANCOVA]], [[MANOVA]], [[MANCOVA]], ordinary [[linear regression]], [[t-test|''t''-test]] and [[F-test|''F''-test]]. The general linear model is a generalization of multiple linear regression model to the case of more than one dependent variable. If '''Y''', '''B''', and '''U''' were [[column vector]]s, the matrix equation above would represent multiple linear regression.
Hypothesis tests with the general linear model can be made in two ways: [[multivariate statistics|multivariate]] or as several independent [[univariate]] tests. In multivariate tests the columns of '''Y''' are tested together, whereas in univariate tests the columns of '''Y''' are tested independently, i.e., as multiple univariate tests with the same design matrix.
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